Asymptotic properties of homogeneity tests

Tests for the homogeneity of samples from the Poisson and Gamma distributions are considered based on the C(α ) procedure of Neyman and on maximum likelihood estimators. These are shown to be equivalent in spite of the fact that the null hypothesis lies on the boundary of the parameter space, which is contrary to the assumptions under which the C(α ) test is derived. Moreover, it is shown that neither procedure has been proved to work unless the disturbing distribution is assumed to have a zero third moment, a so far unexplained phenomenon. Furthermore, it is pointed out that the equivalence of the two types of tests ceases to hold when the null hypothesis involves more than one parameter and at least one of these lies on the boundary of the parameter space.